Can anyone explain to me why CDT two-boxes?

Suppose that Omega is wrong with probability p<1 (this is a perfectly realistic and sensible case). What does (your interpretation of) CDT do in this case, and with what probability?

Here is my EDT calculation:

• calculate p(2box|1box prediction)1001000+p(2box|2box prediction)1000=1001000(1-p)+1000p

• calculate p(1box|1box prediction)1001000+p(1box|2box prediction)1000=1001000p+1000(1-p)

• pick largest of the two (which is 1-box if p < 50%, 2-box if p > 50%).

Thus one should 1-box even if Omega is slightly better than chance.

You are using the wrong sense of "can" in "cannot make different decisions". The every day subjective experience of "free will" isn't caused by your decisions being indeterminate in an objective sense, that's the incoherent concept of libertarian free will. Instead it seems to be based on our decisions being dependent on some sort of internal preference calculation, and the correct sense of "can make different decisions" to use is something like "if the preference calculation had a different outcome that would result in a different decision".

Otherwise results that are entirely random would feel more free than results that are based on your values, habits, likes, memories and other character traits, i. e. the things that make you you. Not at all coincidentally this is also the criterion whether it makes sense to bother thinking about the decision.

You yourself don't know the result of the preference calculation before you run it, otherwise it wouldn't feel like a free decision. But whether Omega knows the result in advance has no impact on that at all.

The reason why you lose in chess is because you will make the wrong moves, and the reason why you will make the wrong moves is that there are much too many of them to make it likely enough that you will find the right ones by chance. This is not the case in a game that consists of only two different moves.

In a game with two moves, you want to model the other person, and play one level higher than that. So if I take the role of Omega and put you in Newcomb's problem, and you think I'll expect you to two box because you've argued in favor of two boxing, then you expect me to put money in only one box, so you want to one box, thereby beating your model of me. But if I expect you to have thought that far, then I want to put money in both boxes, making two boxing the winning move, thereby beating my model of you. And you expect me to have thought that far, you want to play a level above your model of me and one box again.

If humans followed this kind of recursion infinitely, it would never resolve and you couldn't do better than maximum entropy in predicting the other person's decision. But people don't do that, humans tend to follow very few levels of recursion when modeling others (example here, you can look at the comments for the results.) So if one person is significantly better at modeling the other, they'll have an edge and be able to do considerably better than maximum entropy in guessing the other person's choice.

Omega is a hypothetical entity who models the universe perfectly. If you decide to one box, his model of you decides to one box, so he plays a level above that and puts money in both boxes. If you decide to two box, his model of you decides to two box, so he plays a level above that and only puts money in one box. Any method of resolving the dilemma that you apply, his model of you also applies; if you decide to flip a coin, his model of you also decides to flip a coin, and because Omega models the whole universe perfectly, not just you, the coin in his model shows the same face as the coin you actually flip. This does essentially require Omega to be able to fold up the territory and put it in his pocket, but it doesn't require any backwards causality. Real life Newcomblike dilemmas involve predictors who are very reliable, but not completely infallible.

What if you also tell them that you've made a prediction about them, and if your prediction is correct, they will get the money and not be shot even if their decision was to get shot? (If your prediction was wrong, the same happens as in your original game.)

What if you were in that very situation, with Omega, whose predictions are always right, holding the money and the gun? Could you make a distinction between the choices offered to you?

I could choose either, knowing that the results would be the same either way. Either I choose the money, in which case Omega has predicted that I will choose the money, and I get the money and don't get shot, or I choose the bullet, in which case, Omega has predicted that I choose the bullet, and I will get the money and not get shot. In this case, you don't need Omega's perfect prediction to avoid shooting the other person, you can just predict that they'll choose to get shot every time, because whether you're right or wrong they won't get shot, and if you want to shoot them, you should always predict that they'll choose the money, because predicting that they'll choose the money and having them choose the bullet is the only branch that results in shooting them. Similarly, if you're offered the dilemma, you should always pick the money if you don't want to get shot, and the bullet if you do want to get shot. It's a game with a very simple dominant strategy on each side.

If the simulation is correct, a program that chooses to one-box will get $1,000,000, and a program that chooses to two-box will get $1,000. I wouldn't call that "not having a choice".

I meant it assigned a probability p>0 to violation of causality being possible.

It intrinsically says nothing about causality violation. All zero is not a probability and lack of infinite certainty issues are independent of the decision theory. The decision theory just works with whatever your map contains.

Actual Newcomb doesn't include an omniscient being; I quote from Wikipedia:

However, the original discussion by Nozick says only that the Predictor's predictions are "almost certainly" correct, and also specifies that "what you actually decide to do is not part of the explanation of why he made the prediction he made".

Except that this is false, so nevermind.

Also, actual knowledge of everything aside from the Predictor is possible without time travel. It's impossible in practice, but this is a thought experiment. You "just" need to specify the starting position of the system, and the laws operating on it.

If you say that Omega makes its prediction purely based on the past, Newcomb becomes trivial as well.

Omega makes its prediction purely based on the past (and present).

That being the case which decision would you say is trivially correct? Based on what you have said so far I can't predict which way your decision would go.

CDT, then, isn't aware of the payoff matrix. It reasons as follows: Either Omega put money in boxes A and B, or only in box B. If Omega put money in both boxes, I'm better off taking both boxes. If Omega put money only in box B, I should also take both boxes instead of only box A. CDT doesn't deal with the fact that which of these two games it's playing depends on what it will choose to do in each case.

No, this is false. CDT is the one using the standard payoff matrix, and you are the one refusing to use the standard payoff matrix and substituting your own.

In particular: the money is either already there, or not already there. Once the game has begun, the Predictor is powerless to change things.

The standard payoff matrix for Newcomb is therefore as follows:

• Omega predicts you take two boxes, you take two boxes, you get $n>0.
• Omega predicts you take two boxes, you take two boxes, you get 0.
• Omega predicts you take one box, you take one box, you get $m>n.
• Omega predicts you take one box, you take two boxes, you get $m+n>m.

The problem becomes trivial if, as you are doing, you refuse to consider the second and fourth outcomes. However, you are then not playing Newcomb's Problem.

I think people people may be confused by your word choice.

I do disagree with 3, though I disagree (mostly connotatively) with 1 and 2 as well.

The arguments you refer to were not written at the time I wrote my previous response, so I'm not sure what your point in the "Edit" is.

Nevertheless, I'll write my response to your argument now.

In theoretical Newcomb, CDT doesn't care about the rule of Omega being right, so CDT does not play Newcomb.

You are correct when you say that CDT "doesn't care" about Omega being right. But that doesn't mean that CDT agents don't know that Omega is going to be right. If you ask a CDT agent to predict how they will do in the game, they will predict that they will earn far less money than someone who one-boxes. There is no observable fact that a one-boxer and a two-boxer will disagree on (at least in this sense). The only disagreement the two will have is about the counterfactual statement "if you had made a different choice, that box would/would not have contained money".

That counterfactual statement is something that different decision theories implicitly give different views on. Its truth or falsity is not in the problem; it's part of the answer. CDT agents don't rule out the theoretical possibility of a predictor who can accurately predict their actions. CDT just says that the counterfactual which one-boxers use is incorrect. This is wrong, but CDT is just giving a wrong answer to the same question.

This will be my last comment on this thread. I've read Nozick. I've also read much of the current literature on Newcomb's problem. While Omega is sometimes described as a perfect predictor, assuming that Omega is a perfect predictor is not required in order to get an apparently paradoxical result. The reason is that given no backwards causation (more on that below) and as long as Omega is good enough at predicting, CDT and EDT will recommend different decisions. But both approaches are derived from seemingly innocuous assumptions using good reasoning. And that feature -- deriving a contradiction from apparently safe premisses through apparently safe reasoning -- is what makes something a paradox.

Partisans will argue for the correctness or incorrectness of one or the other of the two possible decisions in Newcomb's problem. I have not given any argument. And I'm not going to give one here. For present purposes, I don't care whether one-boxing or two-boxing is the correct decision. All I'm saying is what everyone who works on the problem agrees about. In Newcomb problems, CDT chooses two boxes and that choice has a lower expected value than taking one box. EDT chooses one box, which is strange on its face, since the decision now is presumed to have no causal relevance to the prediction. Yet, EDT recommends the choice with the greater expected value.

The usual story assumes that there is no backwards causation. That is why Nozick asks the reader (in the very passage you quoted, which you really should read more carefully) to: "Suppose we establish or take as given that there is no backwards causality, that what you actually decide to do does not affect what [the predictor] did in the past, that what you actually decide to do is not part of the explanation of why he made the prediction he made." If we don't follow Nozick in making this assumption -- if we assume that there is backwards causation -- CDT does not "reject the case" at all. If there is backwards causation and CDT has that as an input, then CDT will agree with EDT and recommend taking one box. The reason is that in the case of backwards causation, the decision now is causally relevant to the prediction in the past. That is precisely why Nozick ignores backwards causation, and he is utterly explicit about it in the first three sentences of the passage you quoted. So, there is good reason to consider only the case where you know (or believe) that there is no backwards causation because in that case, CDT and EDT paradoxically come apart.

But neither CDT nor EDT excludes any causal structure. CDT and EDT are possible decision theories in worlds with closed time-like curves. They're possible decision theories in worlds that have physical laws that look nothing like our own physical laws. CDT and EDT are theories of decision, not theories of physics or metaphysics.

Both ways of setting it up allow pre-commitment solutions. The fact that one might not have thought about such solutions in time to implement them is not relevant to the question of which decision theory one ought to implement.

Why do you think it matters whether you have heard of the game before or whether you know that Predictor (Omega ... whatever) is looking?

I admit that your way of setting it up makes a Haynes-style realization of the problem unlikely. But so what? My way of setting it up is still a Newcomb problem. It has every bit of logical / decision theoretical force as your way of setting it up. The point of the problem is to test decision theories. CDT (without pre-commitment) is going to take two boxes on your set-up and also on mine. And that should be enough to say that Newcomb problems are possible. (Nomologically possible, not just metaphysically possible or logically possible.)

Can you expand on the difference here?

If I have a "strong enough" precommitment to do whatever gets me the most money, and Predictor privately sets the payoffs for (1box, 2box) such that 1box > 2box, then Predictor can predict that upon being told the payoffs, I will 1box, and Predictor will therefore set the boxes as for the 1box scenario. Conversely, if Predictor privately sets the payoffs such that 2boxing gets me the most money, then Predictor can predict that upon being told the payoffs I'll 2box.

So what thorny issues does this delayed specification of the rules prevent me from sidestepping?

It's not completely clear what "backward causality" (or any causality, outside the typical contexts) means, so maybe it can exist. Better to either ignore the concept in this context (as it doesn't seem relevant) or taboo/clarify it.

The meaning of what Andreas was saying was sufficiently clear. He means "you know, stuff like flipping time travel and changing the goddamn past". Trying to taboo causality and sending everyone off to read Pearl would be a distraction. Possibly a more interesting distraction than another "CDT one boxes! Oh, um.... wait... No, Newcomb's doesn't exist. Err... I mean CDT two boxes and it is right to do so so there!" conversation but not an overwhelmingly relevant one.

If backwards causality cannot exist, would you say that your decision can affect the prediction that Omega made before you made your decision?

No. Both the prediction and my decision came about due to past states of the universe (including my brain). They do not influence each other directly. I still take one box and get $1,000,000 and that is the best possible outcome.o. Both the prediction and my decision came about due to past states of the universe (including my brain). I still take one box and get $1,000,000 and that is the best possible outcome I can expect.